A different strategy for detecting signals of election fraud is to look at the distribution of vote and turnout numbers

Democratic societies are built around the principle of free and fair elections, and that each citizen’s vote should count equally. National elections can be regarded as large-scale social experiments, where people are grouped into usually large numbers of electoral districts and vote according to their preferences. The large number of samples implies statistical consequences for the polling results, which can be used to identify election irregularities. Using a suitable data representation, we find that vote distributions of elections with alleged fraud show a kurtosis substantially exceeding the kurtosis of normal elections, depending on the level of data aggregation. As an example, we show that reported irregularities in recent Russian elections are, indeed, well-explained by systematic ballot stuffing. We develop a parametric model quantifying the extent to which fraudulent mechanisms are present. We formulate a parametric test detecting these statistical properties in election results. Remarkably, this technique produces robust outcomes with respect to the resolution of the data and therefore, allows for cross-country comparisons.

Elections can be seen as large-scale social experiments. A country is segmented into a usually large number of electoral units. Each unit represents a standardized experiment, where each citizen articulates his/her political preference through a ballot. Although elections are one of the central pillars of a fully functioning democratic process, relatively little is known about how election fraud impacts and corrupts the results of these standardized experiments (2, 3).

A different strategy for detecting signals of election fraud is to look at the distribution of vote and turnout numbers, like the strategy in ref. 12. This strategy has been extensively used for the Russian presidential and Duma elections over the last 20 y (13–15). These works focus on the task of detecting two mechanisms, the stuffing of ballot boxes and the reporting of contrived numbers. It has been noted that these mechanisms are able to produce different features of vote and turnout distributions than those features observed in fair elections. For Russian elections between 1996 and 2003, these features were only observed in a relatively small number of electoral units, and they eventually spread and percolated through the entire Russian federation from 2003 onward. According to the work by Myagkov and Ordeshook (14), “[o]nly Kremlin apologists and Putin sycophants argue that Russian elections meet the standards of good democratic practice.” This point was further substantiated with election results from the 2011 Duma and 2012 presidential elections (16–18). Here, it was also observed that ballot stuffing not only changes the shape of vote and turnout distributions but also induces a high correlation between them. Unusually high vote counts tend to co-occur with unusually high turnout numbers.